Badiou.

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thank you for pointing out what nonsense the book on paul was. (for a much more interesting, intelligent and elegant treatment of paul, read agamben.) many people seem to be completely enthralled or overexcited by badiou, for reasons that elude me. in the end he's still only hypocritically playing w/ names. wasn't it poincare who said 'one day mathematics will be able to recover from set theory'?

Spectres of Mark

I don't understand the maths, but Bat certainly does (I'll ask him if he'll post on this thread)... however I just don't understand the disparaging of the magnificent St Paul book... there's no maths/ set theory in that as I recall (perhaps I'm remembering it wrongly)... what is supposed to be the problem with this work?

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Let me answer with counter-questions: where on the about 112 "magnificent" pages of the english language edition does one find anything even resembling an argument? An indication of recognition of the well-known and often discussed difficulties that universalist positions seem to face? where indeed does one find even a criticisable delineation of what is meant by universalism?

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My simplistic answer is: he doesn't. Or more precisely – the foundational metaphor at work here is misleading when it comes to situating the position of set theory within Badiou's theoretical edifice.

The important point is Badiou's equation ontology = mathematics. This is a strictly philosophical proposition, and a thoroughly original one, though Badiou argues that it continues a materialist line of thought that runs through Galileo.

Galileo argued that physics fell into the remit of scientists rather than philosophers. Badiou extends this to metaphysics. Philosophers have failed to answer the ontological question because they have hitherto failed to grasp that it is mathematics that articulates what is expressible about being qua being.

It follows from Badiou's position that philosophers should not compare mathematical concepts to their ready made philosophical ones (eg interminable analytic philosophy debates about the "correct" definition of necessity, possibility, conditionality), but rather "close read" mathematics and derive/extract "metaontological" concepts from it. Badiou's exegesis of apparentenance (belonging, or "membership" – though Badiou avoids this interpretation) is a case in point. It is a metaontological extraction of what's at stake in the epsilon relation.

The most important example of this in L'Etre et l'evenment concerns infinity. Badiou argues, compellingly in my view, that the Cantorian conception of transfinite numbers (aka the aleph hierarchy) represents a thoroughly novel – and superior – understanding of infinity, incommensurable to previous philosophical approaches to the question (all of which, in Badiou's view, treat infinity as the horizon of the finite, and are fatally compromised by theology). Mathematics achieves something that rationalist materialist philosophy has always striven for, namely the desacralisation of infinity.

Now, all the debates about ZFC, hypersets, topoi etc occur within this framework. One can quite legitimately question Badiou's focus on ZFC in L'Etre et l'evenment. That book was written 20 years ago and concerns itself with mathematics that were developed 20 years before that. An awful lot has happened since then, not least the spectacular advances in category theory and the renewed question marks over foundation (AFA, hypersets etc).

But I'd argue that

(i) none of these developments has yet entirely displaced ZFC's position within mathematics. If ontology = mathematics, then the peculiar fact that all of mathematics can be rendered in ZFC has to be acknowledged, and one should expect metaontological analysis of ZFC to produce something fruitful.

(ii) even if many of the formulations in L'Etre et l'evenment do need to be drastically revised in the wake of new mathematical discoveries (and for my money there is something dodgy in Badiou's treatment of foundation), this does not weaken Badiou's fundamental proposition that mathematics = ontology – in fact, it can even be said to develop it further. A case in point is Logique du Monde, in which Badiou reworks much of his previous system in the light of topos theory (and this does not involve "ditching" ZFC as borderpatrol states, but rather developing a complex dialectic between ONTO-logy (the science of being) and onto-LOGY (the science of being-there)).

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As a side note, there's a line in analytic philosophy that says similar things for a while. i remember a discussion with a hardcore mathematical analytic platonist a few years ago
(he has since changed his mind), who complained that apart from those researching large cardinals, nobody would do ontology any more. this school of thought is sometimes said to come from goedel.

bat020 said:

The most important example of this in L'Etre et l'evenment concerns infinity. Badiou argues, compellingly in my view, that the Cantorian conception of transfinite numbers (aka the aleph hierarchy) represents a thoroughly novel – and superior – understanding of infinity, incommensurable to previous philosophical approaches to the question (all of which, in Badiou's view, treat infinity as the horizon of the finite, and are fatally compromised by theology). Mathematics achieves something that rationalist materialist philosophy has always striven for, namely the desacralisation of infinity.

if that's the most important example of badiou's method's yield, it's a poor showing, since this position has been commonplace in the philosophy of mathematics for almost a century now. moreover, set theory does not really clarify what infinity is either. the axiom of infinity stipulates the existence of an infinite set. This initial inifinite set is then treated as a black box (like gravity in newtonian physics) and only some of it's core properties, in particular some but not all of its interaction with membership are axiomatised. this leaves open the question of what precisely this infinity is. and the key problems of ZF-like set theory, like the undeciability of AC or GCH, or the embarrasing fact that the naturals need impredicative definition, come precisely from this underspecification of the initial infinite set. of course one could argue, and that's a position i'd take, that that's alltogether no problem because in final analysis, what something is, is how it interacts with other things, but that's an anti-ontological position. and for the kinds of interactions with set theory we currently care about, the axiomatisation sufficies, i.e. banishes contradiction. it should be noted at this point that this perceived consistency of ZF, which is the reason for the domestication of infinity, is merely an empirical fact. it could change tomorrow. there is no reason to believe in the consistency of ZF other than that nobody has to date found a contradiction.

bat020 said:

If ontology = mathematics, then the peculiar fact that all of mathematics can be rendered in ZFC has to be acknowledged, and one should expect metaontological analysis of ZFC to produce something fruitful.

one could turn this on its head: If ontology = mathematics, then one should expect metaontological analysis of ZFC to produce nothing fruitful outside mathematics.

and indeed, badiou seems unable to forge a convincing connection between his micro analysis of ontology in terms of ZF and his macro-sociological lucubrations. of course that's not in itself problematic, indeed it's a conundrum that all sociological analysis faces for reasons of complexity. I guess the question is: is ZF a good metaphor for sociological analysis?

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if you'll forgive my little rambling here, badiou, in a marie-antoinettean
fashion, is trying to have the cake and eat it himself, too. his 'philosophy'
is a unilateral deferral to set mathematics (yes, i'm aware) in the
philosophiarchal 'foundational metaphor' of ontology of numbers. fascinating. (the spectacle expands as the theoria does.) as borderpatrol said, nothing new emerges here. so forget the generalizability of ZF etc or any paradigmization of theoretical malaise - we want to argue the accuracy and legitimacy of use of mathematics, but not conflate the two (which leads to your i & ii, the argument of infallibility - despite the claim to science). (to me, a logic/metaphor of, for example, supplementarity works much better between the two.) wouldn't it be much more interesting & intellectually worthy to admit & work through mistakes rather than en-joy/ining canonic self-referential politics?

borderpatrol - the underspecification - indeed vagueness - of the infinite is one of its extensions, correct? so the only way, as you said, is to define it over a function, but this does not excise ontology (there's still intensive ontology, cf spinoza).

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I would not say that there is nothing new. on the contrary. Badiou seems to be -- and my understanding is hampered by the fact that none of his main works is available in a language that i can read fluently, without dictionary -- a transcedental philosopher. he claims, to phrase it in a more modern language, that all humans have a module built in that allows them to agree uninamously on the truth of falsity of any given ZFC proof, in other words, humans have a ZFC proof checker. this is an admirably clearcut statement.

it is also clearly related to similar transcendental endevours, like that of Kant, whose Critique of Pure Reason is also in significant parts to do with what it is to be an object, how we synthesize the sensory manifolds into ones. And badious proposal is this: by way of set formation, i.e. through the membership operation only.

I do think that similar conceptions have been proposed before, probably coming from goedel and alonzo church. but i don't think they've been worked out as much as they are in A.B.

My problem with AB as transcendental philosopher (which is different from my judgement of him as sociologist) is that (1) i don't think ZFC like object formation is as basic as he claims it to be, i do think that underlying the human set formation ability is something more geometric, an ability to draw boundaries, something that may have been hinted at in spencer brown's "laws of form"; (2) i don't believe human proof checking abilities are as powerful and as objective as he makes them to be; (3) I don't think that this geometric object forming ability is the only quasi-transcendental human ability; (4) i don't think there are transcendentals, only quasi-transcendentals, i.e. we agree with some high probability on something; (5) a lack of connection with the causing mechanisms for the quasi transcendentals: i.e. the similar neural infrastructure and communication/ observation/ expectation as mechanism for the intersubjective synchronisation of the varied neural infrastructures. Finally, (6) AB's enterprise is too foundational for my liking. By that I mean that i think the quality of a theory is in it's holism: how it link together diverse research efforts, be it biology, sociology, mathematics, transcendental deductions. i don't think one can start: this ZFC thing is the basis of everything.

conquering flesh said:

borderpatrol - the underspecification - indeed vagueness - of the infinite is one of its extensions, correct? so the only way, as you said, is to define it over a function, but this does not excise ontology (there's still intensive ontology, cf spinoza).

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sorry, borderpolice (not patrol)...
i might've been trying to ventriloquize somewhat. in any case, i still argue that, antithetic to the content of badiou's philosophy, his philosophy itself forecloses emergence, including its own, and what we're dealing with here is a clockwork of excesses, precisely political. entrapment by the 'quasi-transcendent' (i believe you mean transcendent instead of transcendental, right?) in the hollow of the unilaterally dual (you mentioned brown, but i don't want to put words in your mouth again) in the guise of immanence. i realize i'm not being so clear as i probably can...
being the advocate's devil, i'm not sure whether connection to neuroscience (at least in its current state) is sufficient or necessary. i agree that a (good) theory must be comprehensive, but for that reason, since we're being somewhat speculative here, i think theory (with all its knee-jerk reflexive hangups) should be ditched in favor of principles.

you said that: 'one could argue, and that's a position i'd take, that that's alltogether no problem because in final analysis, what something is, is how it interacts with other things, but that's an anti-ontological position. and for the kinds of interactions with set theory we currently care about, the axiomatisation sufficies, i.e. banishes contradiction'.

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I've been working through Badiou over the past few months and I have to say I'm surprised: all this rhetoric about a 'return to truth' that gets thrown around in relation to him is a little off the mark. Yes, Badiou refuses to abanon the T-word, but his conception of truth is not as nostalgic as you might suspect. Badiou's claim is that Truth pierces holes in sense. I would argue that he is extending contemporary continental thought's critique of presence, not breaking with it, even though he is deeply skeptical of certain aspects of it.

He also has perhaps the keenest literary sense of any contemporary continental philosopher. He gets Beckett.